- Email: [email protected]
Area load frequency control using fuzzy gain scheduling of PI controllers
Electric
Power
Systems
Rescarch
42 (1997)
ELECTRIC POUJER SYSTErnS RESEClRCH
145-152
Area load frequency control using fuzzy gain scheduling of PI controllers C.S. Chang *, Weihui Fu Department
of Electrical
Enginewing,
National
liniwrsity
of Singapore,
Received
8 November
10 Kent
Ridge
Crescent,
Singapore
119260,
Singapore
1996
Abstract This paper presents a new approach to study the area proportional-integral (PI) controllers. The control scheme zero steady state time error and inadvertent interchange. power system with control deadbands and generation performance of the proposed controller. 0 1997 Elsevier Keywords:
Load
frequency
control;
Gain
scheduling;
Fuzzy
load frequency control (LFC) problem using fuzzy gain scheduling of adopts a formulation for the area control error which always guarantees The proposed control has been designed for a four-area interconnected rate constraints. Simulation results confirm the designed control Science S.A.
logic control
1. Introduction Load-frequency control (LFC) is a very important component in power system operation and control for supplying sufficient and reliable electric power with good quality. The objective of the LFC is to satisfy the following classical requirements in a multi-area interconnected power system [15]: (i) Zero steady-state errors of tie-line exchanges and frequency deviations. (ii) Optimal transient behaviour. (iii) In steady state: the power generation levels should satisfy the optimal dispatch conditions. Many investigations have been reported in the past pertaining to load frequency control of a multi-area interconnected power system. In the literature, some control strategies have been proposed based on classical linear control theory [9,12]. However, because of the inherent characteristics of the changing loads, the operating point of a power system changes continuously during a daily cycle. Thus, a fixed controller may no longer be suitable in all operating conditions. There are some authors who have applied variable structure control [1,2,6,1 l] to make the controller insensitive to system parameters change. However, this method re-
* Corresponding
author.
E-mail:
0378-7796/97/$17.00 Q 1997 Elsevier PII SO378-7796(96)01199-6
[email protected] Science
S.A. All rights
reserved.
quires information on the system states which are very difficult to know completely. In view of this, a new area load frequency controller based on fuzzy gain scheduling of PI controller is proposed in this paper. Gain scheduling is a technique commonly used in designing controller for non-linear systems. Its main advantage is that controller parameters can be changed very quickly in responseto changes in the system dynamics because no parameter estimation is required. Besides being an effective method to compensate for non-linear and other predictable variations
in the
system
dynamics,
it
is also
simpler
to
implement than automatic tuning or adaptation. However. conventional gain scheduling also has its drawbacks. One drawback is that the system parameter change may be rather abrupt across the regional boundaries, which may result in unsatisfactory or even unstable performance across the transition regions. Another problem is that accurate linear time-invariant models at various operating points may be too difficult or even impossible to obtain. In order to solve the above-mentioned problems of conventional gain scheduling, this paper introduces a fuzzy rule-based scheme for gain scheduling of PI controllers. Interest in fuzzy logic has grown considerably over the past few years. The fuzzy reasoning approach is motivated by the following advantages [3]: (a) it provides an efficient way of coping with imperfect
C.S. Chang,
146
W. Fu /Electric
Power
information, especially imprecision in available knowledge; (b) it offers flexibility in decision-making processes; and (c) it gives an interesting man/machine interface by simplifying rule extraction from human experts and by allowing a simpler a posteriori interpretation of the system reasoning. In this paper we utilize fuzzy rules and reasoning to determine the controller parameters. Thus the controller has the ability to change its parameters when the system dynamics or the characteristics of the disturbance are changing. Moreover, this method does not need an accurate model of the dynamics of the system under control. To further improve the performance of the controller, a new formulation of the area control error [5] is also adopted. With this method, interconnected systems can always obtain zero steady state time error and inadvertent interchange. The paper is organized as follows: in Section 2, a comprehensive mathematical model of a four-area interconnected power system including governor deadbands and generation rate constraints is presented. A proposed fuzzy gain scheduling controller is described in Section 3. Results obtained from the application are presented in Section 4. The concluding remarks are contained in Section 5.
Systems
Research
42 (1997)
Fig.
145-152
2. The scheme
of fuzzy
gain scheduling.
interchange. The modified expression for area control for the mth area ACEN, is: ACEN,
= AP,,, m+ B,,, AF,, + a,&, + %I,,, = ACE, + a,&, + a,I,
(1)
This equation is different from the conventional area control error since E,,, and Z, are, respectively, the time error and inadvertent interchange of area m and are updated at every sampling instant. ACEN, may be rewritten as ACEN.,.=AP,,,+B..AF;,+z,,(I.,+~i.)
(2)
Setting x,/c(, = 608, [5], then ACEN,
= AP,i, m + B, AF,
+G,s(Apt,, m+4nAFm) dt
2. System model An interconnected power system can be considered as being divided into control areas which are connected by tie lines. In each control area, all generators are assumed to form a coherent group. The power system is subjected to local variations of random magnitude and duration. Hence, it is required to control the deviations of frequency and tie-line power of each control area. The study system, consisting of four areas, is shown in Fig. 1. The power stations are assumed to contain three reheat turbine type thermal units and one hydro unit. The detailed block diagram of this system and data are shown in Appendix A. Ref. [5] proposed a new technique for co-ordinating system-wide corrections of time error and inadvertent
= ACE, + CI,
ACE,
dt
(3)
s
where 1 Em= 60
AF,,, dt,
I, =
s
s
APti, m dt
Thus, the new area control error ACEN, is the sum of the conventional ACE and the integral of the conventional ACE. It will guarantee zero steady state time error and inadvertent interchange. Then, the control vector in continuous mode can be given as U,(t)
= - K,ACEN,(t)
NB
NM
- K,
NS
ZO
s
ACEN,(t)
PS
PM
dt
PB
X = ACENn or AACENn Fig.
1. Simplified
diagram
of an interconnected
system
Fig. 3. Membership
functions
for ACEN,(k)
and AACEN,(k).
(4)
C.S. Chang,
W. Fu /Electric
Power
Systems
Research
Table Fuzzy
Big
1 tuning
147
rules for Kb
NB NMS NS ZO PS PMS PB
ACEN,(k)
x = I$,’ or K,’ functions
145-152
AACEN,(k)
I
Fig. 4. Membership
42 (1997)
NB
NM
NS
ZO
PS
PM
PB
B
B B S S S B B
B B B S B B B
B B B S B B B
B B B S B B B
B B S S S B B
B S S S S S B
S S S B
for KD and K;
The function approach described is used to incorporate governor deadband nonlinearity [13]. It has been found that the backlash nonlinearity tends to produce a continuous sinusoidal oscillation with a natural period of about 2 s. An approximate Fourier series solution has been developed as follows [13]: F(x, i) = 0.8x - g i 71 where the above Fourier coefficients stand for a backlash of 0.05%. In practice, there is a maximum limit to the rate of change in generation power of a steam plant [14]. A typical value of the generation rate constraint for a large reheat-type thermal unit is considered to be 3%/ min, while for hydro units a typical value for raising the generation rate would be 4.5%/s (270%/min) and 6%/s (360%/min) for lowering the generation rate [8]. Here, a value of 3%/min is used for the reheat-type thermal units and 4.5%/s for the hydro unit. 3. The proposed fuzzy gain scheduling controller Gain scheduling is an effective way of controlling systems whose dynamics change non-linearly with operating conditions [4]. It is normally used when the relationship between the system dynamics and operating conditions are known, and for which a single linear time-invariant model is insufficient. In this paper, we use this technique to schedule the parameters of the PI controller according to change of the new area control error ACEN, and AACEN, as depicted in Fig. 2.
In the proposed scheme, PI parameters are determined based on the current ACEN, and its first difference AACEN,. It is also assumed that &., and Ki are in
the prescribedranges[Kp,min,&,,,J and [Kl,rnin,~1,,,,1, respectively. The appropriate ranges are determined experimentally for each area. For convenience, K,, and K1 are normalized into a range between zero and one by the following linear transformation:
The parameters Kb and K; are determined fuzzy rules of the form if ACEN,(k)
is Ai and AACEN,(k)
then Kb is C, and K; is Di
by a set of
is Bi,
i = 1,2 ,..., n.
(7)
Here, A, B,, C, and Di are fuzzy sets on the corresponding supporting sets. The membership function (MF) sets for ACEN,(k) and AACEN,(k) are shown in Fig. 3, in which N, P, ZO, S, M, B, NB and NM represent respectively negative, positive, approximately zero, small, medium, big, negative big, and negative medium. The fuzzy sets C, and Di can be either Big or Small and are characterized by the membership functions shown in Fig. 4, where the grade of the membership functions p and the variable x ( = Kb or K;) have the following relation [4]: Table Fuzzy
2 tuning
rules for K; AACEN,(k)
ACEN,(k) 1
al Fig. 5. Example
of a desired
step response.
NB NMS NS ZO PS PMS PB
NB
NM
NS
ZO
PS
PM
PB
S
S S B B B S S
S B B B B B S
S B B B B B S
S B B B B B S
S S B B B S S
S S S B S S S
S B S S
C.S. Chang,
148
W. FM /Electric
Power
Systems
0
2
-0.04
-10. :‘-
ij-q" -0.06
jr-- -15
0
50
100
in area
1: (a) AF,;
psrnall(~) = - t In x or xsma&)
(b) AF2; (c) AF,;
= 1 - e -‘+
(d) BP,,, ,; (e) E,; (f) I,. (-.
for Big (8)
The fuzzy rules in Eq. (7) may be extracted by experience. Here we derive the rules experimentally based on the step response of the process. Fig. 5 shows an example of a desired time response. At the beginning, i.e., around a,, we need a big control signal, so the PI controller should have a large proportional gain to improve the system response, but a small integral gain to prevent overshoot [7]. Therefore, the rule around a, should be if ACEN,(k)
is NB and AACEN,(k)
is ZO,
then Kb is Big and K; is Small. Around point a2 in Fig. 5 we need a small control signal, so the PI controller should have a small proportional gain to reduce to system oscillations, but a big integral gain to eliminate the steady-state error. Then, the following fuzzy rule is taken. if ACEN,(k)
is ZO and AACEN,(K)
50 (0
= e ~ 4V for Small
pgig(x) = - a ln(1 - X) or X,&L)
0
Sec.
(e)
increase
145-152
-5..
r-----l
6. 1% step load
42 (1997)
O-
~-0.02
Fig.
Research
100 sec.
proposed
controller;
..., fixed
PI controller.)
integral gain K; respectively, in which B stands for Big, and S for Small. The value of the ith rule in Eq. (7) ,uuiis obtained by the product of the MF values of ACEN,(k) and AACEN,(k):
pi = .P~[ACWA~)I. P~WACW,,(~)I
(9)
where pJACEN,(k)] is the MF value of the fuzzy set Ai according to the value of ACEN,(k), and pi[AACEN,(k)] the MF value of the fuzzy set Bj according to the value of AACEN,(k). Based on ,ui, the values of Kb and K; for each rule are determined from their corresponding membership functions. From the membership functions in Fig. 3, we can see that:
iI,= Then, defuzzification
(10) yields the following:
is PS,
then Kb is Small and K; is Big. Thus, a set of rules, as shown in Tables 1 and 2, may be used to adapt the proportional gain Kj, and the
(11) where Kp,i is the value of Kb corresponding to the grade ,U~for the ith rule, and K;,, is obtained in the same way.
C.S. Chang,
W. Fu /Electric
Power
Systems
-0.1
Research
42 (1997)
145-152
149
-0.1 0
50 (a)
100
0
50
Sec.
100 Sec.
(b)
5x1o-3
3 a
i d -5-F I
\ -10
-10 0
50 cc>
Fig. 7. 1% step load increase
in areas
1 and 3: (a) AF,;
100
(12)
4. Results and analysis In order to compare the performance of the proposed controllers, three different kinds of perturbation have been considered: 1. 1% step load increase in area 1 [Fig. 6(a)-(f)]. 2. 1% step load increase in area 1 and 3 simultaneously [Fig. 7(a)-(d)]. 3. 1% step load increase in area 1 and 3, 1% step load decrease in area 2 simultaneously [Fig. 8(a)-(h)]. For comparison, system responses with a PI controller of fixed gains are also shown. The PI controller’s fixed gains are optimized using a method as illustrated in Appendix B [8]. It is observed that the setting time of the proposed control strategy is much shorter than that with a fixed PI controller, and with the proposed controller a large overshoot also can be avoided. Moreover, the proposed controllers exhibit robustness as they show superior performance in all three different kinds of perturbation. Due to the use of the modified ACEN (Eq. (3)), all the controllers lead to zero steady-state frequency and tie-line power deviations. The proposed control has also achieved for each area zero steady values for the time error and inadvertent interchange.
100 Sec.
(4
(b) AF3; (c) Pt,, ,; (d) Pt,, ,,. (-,
- Kp,min)Kb + Kp,min
KI = W,,max - K~,mirJKi + K~,tin
50
Sec.
After obtaining K;, and K;, the PI controller parameters are calculated from the following equations: Kp = W,,m
0
proposed
controller;
..., fixed
PI controller.)
5. Conclusions An alternative method of large-scale interconnected power system load frequency control using fuzzy gain scheduling has been proposed in this paper. Through simulations, the performance of the proposed controllers is shown to compare favourably with that of optimum fixed-gain controllers. The proposed algorithm is very simple, effective and robust. In practice, it can be implemented with few changes to the existing controller configurations.
Appendix A. Simulated system data In the following, most of the parameters of the four-area study system in Fig. 9 are from Refs. [10,16] and some of the parameters have been modified: Reheat turbine type thermal units:
R = 2.4 Hz/p.u., T,, = 0.3 s, Hydro
Tg = 0.2,
Kp = 120 Hz/p.u.,
T,, = 20 s,
T, = 20 s.
unit:
R= 2.4 Hz/p.u., T, = 10 s, Synchronizing
T,,=
K,, = 0.333,
T, =48.7 s, T,=
ls,
Kp = 80 Hz/p.u.,
constants:
T,, = T14= Tz3 = 0.0707
Frequency
bias constants:
T, =0.513 s, Tp= 13 s.
150
C.S. Chang,
0.05,
k
0 5 d -0.05
-0.1
0
W. Fu / Electric
Power
Research
0.05
Oi d
-0.05 -0.1’ 50 (0)
0
100 Set
42 (1997)
145-152
:k’ :”
’
I 100
50
(b)
0.1 ,
-0.1
Systems
Sec.
1
’ 0
I 105
50 Cc)
0.02
-151 0
SK.
50 (d)
100 5.32.
, I 0
i d
; d
-5 -10
-0.01
’ 0
I 50 (e)
100
Sec.
0
-0.1 i d -0.2 L7
(h) Fig. 8. 1% step load increase in areas 1 and 3, 1% step load decrease (h) I,. (-, proposed controller; -, fixed PI controller.)
B, = B, = B, = B4 = 0.425
Inadvertent
interchange
bias constants:
aI = a2 = a3 = a, = 0.001 Appendix
B. Optimum
A performance J=
~ k=O
[AF?(k)
PI gain setting
index + AP~i, i(k)]
(BII. 1)
in area 2: (a) AF,;
(b) AF2; (c) AF,;
SK.
(d) AP,,
,; (e) AP,,,>;
(f) AP,,,,;
(g) E,;
is used for obtaining the optimum PI gain setting. While optimizing the PI gain setting of area 1, areas 2, 3 and 4 are considered uncontrolled and a step load perturbation of 1% of the nominal load is considered in area 1. Fig. 10 shows J=f(K,,) for several values of K,, for a step load perturbation in area 1. K,, = 0.08 and K,, = 0.02 are found to be the optimum settings. Following the same procedure the optimum values of PI controllers are obtained as follows:
C.S. Chang,
W. Fu/Electric
Fig. 9. Detailed
Power
block
Systems
diagram
Research
of the studied
42 (1997)
145-1.52
151
system.
J 0.115
/
0.11
/
0.105
/ / 0.1
0.095
0
0.005
,
I
/
I
0.01
0.015
0.02
0.025
I
0.03
0.035
0 .04
J&I
Fig. 10. Performance index K,,, = 0.10; 7, Kp, = 0.12.
J =f(K,,)
for
various
values
of K,,:
I, I&, = 0.0; 2, Kp, = 0.02;
3, Kp, = 0.04;
4, Kp, = 0.06;
5, Kp, = 0.08;
Appendix C. Nomenclature
Area
KP
K
1 2
0.08 0.1 0.1
0.02 0.02
ACE, ACENi Apt, i
0.02 0.06
AF,
3 4
0.2
Bi
area control error of area i new area control error of area i incremental change in aggregate tie-line power of area i incremental frequency change of area i frequency bias constant of area i
6,
152
C.S. Chang,
I,
Tri T,,
Tz,
T3
TW R,
AP,i APc,
i
W. Fu /Electric
Power
time error bias setting of area i time error of area i inadvertent interchange bias constant of area i inadvertent interchange accumulation of area i gain constant of area i time constant of area i time constants of the governor of area i time constants of the turbine of area i reheat coefficient of the steam turbine of area i reheat time constant of area i time constants of hydrogovernor water starting time static speed drop of the uncontrolled turbine generator of area i load disturbance of area i control signal of area i area index (1, 2, 3, or 4)
References [1] T. Kennedy, SM. Hoyt and C.F. Abell. Variable, non-linear tie-line frequency bias for interconnected systems control, IEEE Trans. Power Syst., 3 (3) (1988). [2] Z.M. Ai-Hamouz and Y.L. Abdel-Magid, Variable structure load frequency controllers for multiarea power systems, Int. J. Electr. Power EnergJa Syst., 15 (5) (1993). [3] CC. Lee, Fuzzy logic in control systems: Fuzzy logic controller, parts I and II, IEEE Trans. Sq’st., Man Cybern.. 20 (2) (1990). [4] Zhen-Yu Zhao, Masayoshi Tomizuka and Satoru Isaka, Fuzzy
Systems
Research
42 (1997)
145-152
gain scheduling of a PID controllers, IEEE Trans. Syst., Man Cybern., 23 (5) (1993). J. Nanda, D.P. Kothari and D. Das, Discrete151 M.L. Kothari, mode automatic generation control of a two-area reheat thermal system with new area control error, IEEE Trans. Power Syst., 4 (2) (1989). [61 A. Kumar, O.P. Malik and G.S. Hope, Variable-structure-system control applied to AGC of an interconnected power system, IEE Proc., 132 Pt. C (I) (1985). M.L. Kothari and P.S. Satsangi, Automatic genera171 .I. Nanda, tion control of an interconnected hydrothermal system in continuous and discrete modes considering generation rate constraints, IEE Proc., 130 Pt. D, (1) (1983). PI D. Das, J. Nanda, M.L. Kothari and D.P. Kothari, Automatic generation control of a hydrothermal system with new area control error considering generation rate constraint, Electr. Mach. Power Syst., 18 (1990) 461-471. and M.H. Hamza, Comparison of three al[91 P. Agathoklis gorithms for load frequency control, Electr. Power Syst. Res., 7 (1984) 1655172. [lOI 0. P. Malik, A. Kumar and G.S. Hope, A load frequency control algorithm based on a generalized approach. IEEE Trans. Power Syst., 3 (2) (1988). [“I A. Rubaai and V. Udo. An adaptive control scheme for load-frequency control of multiarea power systems. Parts I and II. Electr. Power Syst. Res., 24 (1982). Regulation error in load frequency u-4 A. Bose and 1. Atiyyah, control, fEEE Trans. Power Appar. Syst., PAS-99 (2) (1980). G.S. Hope and O.P. Malik, Optimisation of 1131 S.C. Tripathy, load-frequency control parameters for power systems with reheat steam turbines and governor deadband nonlinearity, IEE Proc., 129 Pt. C, (1) (1982). and A. Marched, Dynamic behaviour of AGC u41 E.B. Shahrodi systems including the effects of nonlinearities, IEEE Trans. Power Syst.. PAS-104 (12) (1985). D.N. Ewart, L.H. Fink and A.G. USI N. Jaleey, L.S. VanSlyck, Hoffman, Understanding automatic generation control, IEEE Trans. Power Syst., PAS-89 (4) (1990). Lu, Chun-Chang Liu and Chi-Jui Wu, Effect of U61 Chun-Feng battery energy storage system on load frequency control considering governor deadband and generation rate constraint, IEEE Trans. Power Syst, IO (3) (1995).
Power
Systems
Rescarch
42 (1997)
ELECTRIC POUJER SYSTErnS RESEClRCH
145-152
Area load frequency control using fuzzy gain scheduling of PI controllers C.S. Chang *, Weihui Fu Department
of Electrical
Enginewing,
National
liniwrsity
of Singapore,
Received
8 November
10 Kent
Ridge
Crescent,
Singapore
119260,
Singapore
1996
Abstract This paper presents a new approach to study the area proportional-integral (PI) controllers. The control scheme zero steady state time error and inadvertent interchange. power system with control deadbands and generation performance of the proposed controller. 0 1997 Elsevier Keywords:
Load
frequency
control;
Gain
scheduling;
Fuzzy
load frequency control (LFC) problem using fuzzy gain scheduling of adopts a formulation for the area control error which always guarantees The proposed control has been designed for a four-area interconnected rate constraints. Simulation results confirm the designed control Science S.A.
logic control
1. Introduction Load-frequency control (LFC) is a very important component in power system operation and control for supplying sufficient and reliable electric power with good quality. The objective of the LFC is to satisfy the following classical requirements in a multi-area interconnected power system [15]: (i) Zero steady-state errors of tie-line exchanges and frequency deviations. (ii) Optimal transient behaviour. (iii) In steady state: the power generation levels should satisfy the optimal dispatch conditions. Many investigations have been reported in the past pertaining to load frequency control of a multi-area interconnected power system. In the literature, some control strategies have been proposed based on classical linear control theory [9,12]. However, because of the inherent characteristics of the changing loads, the operating point of a power system changes continuously during a daily cycle. Thus, a fixed controller may no longer be suitable in all operating conditions. There are some authors who have applied variable structure control [1,2,6,1 l] to make the controller insensitive to system parameters change. However, this method re-
* Corresponding
author.
E-mail:
0378-7796/97/$17.00 Q 1997 Elsevier PII SO378-7796(96)01199-6
[email protected] Science
S.A. All rights
reserved.
quires information on the system states which are very difficult to know completely. In view of this, a new area load frequency controller based on fuzzy gain scheduling of PI controller is proposed in this paper. Gain scheduling is a technique commonly used in designing controller for non-linear systems. Its main advantage is that controller parameters can be changed very quickly in responseto changes in the system dynamics because no parameter estimation is required. Besides being an effective method to compensate for non-linear and other predictable variations
in the
system
dynamics,
it
is also
simpler
to
implement than automatic tuning or adaptation. However. conventional gain scheduling also has its drawbacks. One drawback is that the system parameter change may be rather abrupt across the regional boundaries, which may result in unsatisfactory or even unstable performance across the transition regions. Another problem is that accurate linear time-invariant models at various operating points may be too difficult or even impossible to obtain. In order to solve the above-mentioned problems of conventional gain scheduling, this paper introduces a fuzzy rule-based scheme for gain scheduling of PI controllers. Interest in fuzzy logic has grown considerably over the past few years. The fuzzy reasoning approach is motivated by the following advantages [3]: (a) it provides an efficient way of coping with imperfect
C.S. Chang,
146
W. Fu /Electric
Power
information, especially imprecision in available knowledge; (b) it offers flexibility in decision-making processes; and (c) it gives an interesting man/machine interface by simplifying rule extraction from human experts and by allowing a simpler a posteriori interpretation of the system reasoning. In this paper we utilize fuzzy rules and reasoning to determine the controller parameters. Thus the controller has the ability to change its parameters when the system dynamics or the characteristics of the disturbance are changing. Moreover, this method does not need an accurate model of the dynamics of the system under control. To further improve the performance of the controller, a new formulation of the area control error [5] is also adopted. With this method, interconnected systems can always obtain zero steady state time error and inadvertent interchange. The paper is organized as follows: in Section 2, a comprehensive mathematical model of a four-area interconnected power system including governor deadbands and generation rate constraints is presented. A proposed fuzzy gain scheduling controller is described in Section 3. Results obtained from the application are presented in Section 4. The concluding remarks are contained in Section 5.
Systems
Research
42 (1997)
Fig.
145-152
2. The scheme
of fuzzy
gain scheduling.
interchange. The modified expression for area control for the mth area ACEN, is: ACEN,
= AP,,, m+ B,,, AF,, + a,&, + %I,,, = ACE, + a,&, + a,I,
(1)
This equation is different from the conventional area control error since E,,, and Z, are, respectively, the time error and inadvertent interchange of area m and are updated at every sampling instant. ACEN, may be rewritten as ACEN.,.=AP,,,+B..AF;,+z,,(I.,+~i.)
(2)
Setting x,/c(, = 608, [5], then ACEN,
= AP,i, m + B, AF,
+G,s(Apt,, m+4nAFm) dt
2. System model An interconnected power system can be considered as being divided into control areas which are connected by tie lines. In each control area, all generators are assumed to form a coherent group. The power system is subjected to local variations of random magnitude and duration. Hence, it is required to control the deviations of frequency and tie-line power of each control area. The study system, consisting of four areas, is shown in Fig. 1. The power stations are assumed to contain three reheat turbine type thermal units and one hydro unit. The detailed block diagram of this system and data are shown in Appendix A. Ref. [5] proposed a new technique for co-ordinating system-wide corrections of time error and inadvertent
= ACE, + CI,
ACE,
dt
(3)
s
where 1 Em= 60
AF,,, dt,
I, =
s
s
APti, m dt
Thus, the new area control error ACEN, is the sum of the conventional ACE and the integral of the conventional ACE. It will guarantee zero steady state time error and inadvertent interchange. Then, the control vector in continuous mode can be given as U,(t)
= - K,ACEN,(t)
NB
NM
- K,
NS
ZO
s
ACEN,(t)
PS
PM
dt
PB
X = ACENn or AACENn Fig.
1. Simplified
diagram
of an interconnected
system
Fig. 3. Membership
functions
for ACEN,(k)
and AACEN,(k).
(4)
C.S. Chang,
W. Fu /Electric
Power
Systems
Research
Table Fuzzy
Big
1 tuning
147
rules for Kb
NB NMS NS ZO PS PMS PB
ACEN,(k)
x = I$,’ or K,’ functions
145-152
AACEN,(k)
I
Fig. 4. Membership
42 (1997)
NB
NM
NS
ZO
PS
PM
PB
B
B B S S S B B
B B B S B B B
B B B S B B B
B B B S B B B
B B S S S B B
B S S S S S B
S S S B
for KD and K;
The function approach described is used to incorporate governor deadband nonlinearity [13]. It has been found that the backlash nonlinearity tends to produce a continuous sinusoidal oscillation with a natural period of about 2 s. An approximate Fourier series solution has been developed as follows [13]: F(x, i) = 0.8x - g i 71 where the above Fourier coefficients stand for a backlash of 0.05%. In practice, there is a maximum limit to the rate of change in generation power of a steam plant [14]. A typical value of the generation rate constraint for a large reheat-type thermal unit is considered to be 3%/ min, while for hydro units a typical value for raising the generation rate would be 4.5%/s (270%/min) and 6%/s (360%/min) for lowering the generation rate [8]. Here, a value of 3%/min is used for the reheat-type thermal units and 4.5%/s for the hydro unit. 3. The proposed fuzzy gain scheduling controller Gain scheduling is an effective way of controlling systems whose dynamics change non-linearly with operating conditions [4]. It is normally used when the relationship between the system dynamics and operating conditions are known, and for which a single linear time-invariant model is insufficient. In this paper, we use this technique to schedule the parameters of the PI controller according to change of the new area control error ACEN, and AACEN, as depicted in Fig. 2.
In the proposed scheme, PI parameters are determined based on the current ACEN, and its first difference AACEN,. It is also assumed that &., and Ki are in
the prescribedranges[Kp,min,&,,,J and [Kl,rnin,~1,,,,1, respectively. The appropriate ranges are determined experimentally for each area. For convenience, K,, and K1 are normalized into a range between zero and one by the following linear transformation:
The parameters Kb and K; are determined fuzzy rules of the form if ACEN,(k)
is Ai and AACEN,(k)
then Kb is C, and K; is Di
by a set of
is Bi,
i = 1,2 ,..., n.
(7)
Here, A, B,, C, and Di are fuzzy sets on the corresponding supporting sets. The membership function (MF) sets for ACEN,(k) and AACEN,(k) are shown in Fig. 3, in which N, P, ZO, S, M, B, NB and NM represent respectively negative, positive, approximately zero, small, medium, big, negative big, and negative medium. The fuzzy sets C, and Di can be either Big or Small and are characterized by the membership functions shown in Fig. 4, where the grade of the membership functions p and the variable x ( = Kb or K;) have the following relation [4]: Table Fuzzy
2 tuning
rules for K; AACEN,(k)
ACEN,(k) 1
al Fig. 5. Example
of a desired
step response.
NB NMS NS ZO PS PMS PB
NB
NM
NS
ZO
PS
PM
PB
S
S S B B B S S
S B B B B B S
S B B B B B S
S B B B B B S
S S B B B S S
S S S B S S S
S B S S
C.S. Chang,
148
W. FM /Electric
Power
Systems
0
2
-0.04
-10. :‘-
ij-q" -0.06
jr-- -15
0
50
100
in area
1: (a) AF,;
psrnall(~) = - t In x or xsma&)
(b) AF2; (c) AF,;
= 1 - e -‘+
(d) BP,,, ,; (e) E,; (f) I,. (-.
for Big (8)
The fuzzy rules in Eq. (7) may be extracted by experience. Here we derive the rules experimentally based on the step response of the process. Fig. 5 shows an example of a desired time response. At the beginning, i.e., around a,, we need a big control signal, so the PI controller should have a large proportional gain to improve the system response, but a small integral gain to prevent overshoot [7]. Therefore, the rule around a, should be if ACEN,(k)
is NB and AACEN,(k)
is ZO,
then Kb is Big and K; is Small. Around point a2 in Fig. 5 we need a small control signal, so the PI controller should have a small proportional gain to reduce to system oscillations, but a big integral gain to eliminate the steady-state error. Then, the following fuzzy rule is taken. if ACEN,(k)
is ZO and AACEN,(K)
50 (0
= e ~ 4V for Small
pgig(x) = - a ln(1 - X) or X,&L)
0
Sec.
(e)
increase
145-152
-5..
r-----l
6. 1% step load
42 (1997)
O-
~-0.02
Fig.
Research
100 sec.
proposed
controller;
..., fixed
PI controller.)
integral gain K; respectively, in which B stands for Big, and S for Small. The value of the ith rule in Eq. (7) ,uuiis obtained by the product of the MF values of ACEN,(k) and AACEN,(k):
pi = .P~[ACWA~)I. P~WACW,,(~)I
(9)
where pJACEN,(k)] is the MF value of the fuzzy set Ai according to the value of ACEN,(k), and pi[AACEN,(k)] the MF value of the fuzzy set Bj according to the value of AACEN,(k). Based on ,ui, the values of Kb and K; for each rule are determined from their corresponding membership functions. From the membership functions in Fig. 3, we can see that:
iI,= Then, defuzzification
(10) yields the following:
is PS,
then Kb is Small and K; is Big. Thus, a set of rules, as shown in Tables 1 and 2, may be used to adapt the proportional gain Kj, and the
(11) where Kp,i is the value of Kb corresponding to the grade ,U~for the ith rule, and K;,, is obtained in the same way.
C.S. Chang,
W. Fu /Electric
Power
Systems
-0.1
Research
42 (1997)
145-152
149
-0.1 0
50 (a)
100
0
50
Sec.
100 Sec.
(b)
5x1o-3
3 a
i d -5-F I
\ -10
-10 0
50 cc>
Fig. 7. 1% step load increase
in areas
1 and 3: (a) AF,;
100
(12)
4. Results and analysis In order to compare the performance of the proposed controllers, three different kinds of perturbation have been considered: 1. 1% step load increase in area 1 [Fig. 6(a)-(f)]. 2. 1% step load increase in area 1 and 3 simultaneously [Fig. 7(a)-(d)]. 3. 1% step load increase in area 1 and 3, 1% step load decrease in area 2 simultaneously [Fig. 8(a)-(h)]. For comparison, system responses with a PI controller of fixed gains are also shown. The PI controller’s fixed gains are optimized using a method as illustrated in Appendix B [8]. It is observed that the setting time of the proposed control strategy is much shorter than that with a fixed PI controller, and with the proposed controller a large overshoot also can be avoided. Moreover, the proposed controllers exhibit robustness as they show superior performance in all three different kinds of perturbation. Due to the use of the modified ACEN (Eq. (3)), all the controllers lead to zero steady-state frequency and tie-line power deviations. The proposed control has also achieved for each area zero steady values for the time error and inadvertent interchange.
100 Sec.
(4
(b) AF3; (c) Pt,, ,; (d) Pt,, ,,. (-,
- Kp,min)Kb + Kp,min
KI = W,,max - K~,mirJKi + K~,tin
50
Sec.
After obtaining K;, and K;, the PI controller parameters are calculated from the following equations: Kp = W,,m
0
proposed
controller;
..., fixed
PI controller.)
5. Conclusions An alternative method of large-scale interconnected power system load frequency control using fuzzy gain scheduling has been proposed in this paper. Through simulations, the performance of the proposed controllers is shown to compare favourably with that of optimum fixed-gain controllers. The proposed algorithm is very simple, effective and robust. In practice, it can be implemented with few changes to the existing controller configurations.
Appendix A. Simulated system data In the following, most of the parameters of the four-area study system in Fig. 9 are from Refs. [10,16] and some of the parameters have been modified: Reheat turbine type thermal units:
R = 2.4 Hz/p.u., T,, = 0.3 s, Hydro
Tg = 0.2,
Kp = 120 Hz/p.u.,
T,, = 20 s,
T, = 20 s.
unit:
R= 2.4 Hz/p.u., T, = 10 s, Synchronizing
T,,=
K,, = 0.333,
T, =48.7 s, T,=
ls,
Kp = 80 Hz/p.u.,
constants:
T,, = T14= Tz3 = 0.0707
Frequency
bias constants:
T, =0.513 s, Tp= 13 s.
150
C.S. Chang,
0.05,
k
0 5 d -0.05
-0.1
0
W. Fu / Electric
Power
Research
0.05
Oi d
-0.05 -0.1’ 50 (0)
0
100 Set
42 (1997)
145-152
:k’ :”
’
I 100
50
(b)
0.1 ,
-0.1
Systems
Sec.
1
’ 0
I 105
50 Cc)
0.02
-151 0
SK.
50 (d)
100 5.32.
, I 0
i d
; d
-5 -10
-0.01
’ 0
I 50 (e)
100
Sec.
0
-0.1 i d -0.2 L7
(h) Fig. 8. 1% step load increase in areas 1 and 3, 1% step load decrease (h) I,. (-, proposed controller; -, fixed PI controller.)
B, = B, = B, = B4 = 0.425
Inadvertent
interchange
bias constants:
aI = a2 = a3 = a, = 0.001 Appendix
B. Optimum
A performance J=
~ k=O
[AF?(k)
PI gain setting
index + AP~i, i(k)]
(BII. 1)
in area 2: (a) AF,;
(b) AF2; (c) AF,;
SK.
(d) AP,,
,; (e) AP,,,>;
(f) AP,,,,;
(g) E,;
is used for obtaining the optimum PI gain setting. While optimizing the PI gain setting of area 1, areas 2, 3 and 4 are considered uncontrolled and a step load perturbation of 1% of the nominal load is considered in area 1. Fig. 10 shows J=f(K,,) for several values of K,, for a step load perturbation in area 1. K,, = 0.08 and K,, = 0.02 are found to be the optimum settings. Following the same procedure the optimum values of PI controllers are obtained as follows:
C.S. Chang,
W. Fu/Electric
Fig. 9. Detailed
Power
block
Systems
diagram
Research
of the studied
42 (1997)
145-1.52
151
system.
J 0.115
/
0.11
/
0.105
/ / 0.1
0.095
0
0.005
,
I
/
I
0.01
0.015
0.02
0.025
I
0.03
0.035
0 .04
J&I
Fig. 10. Performance index K,,, = 0.10; 7, Kp, = 0.12.
J =f(K,,)
for
various
values
of K,,:
I, I&, = 0.0; 2, Kp, = 0.02;
3, Kp, = 0.04;
4, Kp, = 0.06;
5, Kp, = 0.08;
Appendix C. Nomenclature
Area
KP
K
1 2
0.08 0.1 0.1
0.02 0.02
ACE, ACENi Apt, i
0.02 0.06
AF,
3 4
0.2
Bi
area control error of area i new area control error of area i incremental change in aggregate tie-line power of area i incremental frequency change of area i frequency bias constant of area i
6,
152
C.S. Chang,
I,
Tri T,,
Tz,
T3
TW R,
AP,i APc,
i
W. Fu /Electric
Power
time error bias setting of area i time error of area i inadvertent interchange bias constant of area i inadvertent interchange accumulation of area i gain constant of area i time constant of area i time constants of the governor of area i time constants of the turbine of area i reheat coefficient of the steam turbine of area i reheat time constant of area i time constants of hydrogovernor water starting time static speed drop of the uncontrolled turbine generator of area i load disturbance of area i control signal of area i area index (1, 2, 3, or 4)
References [1] T. Kennedy, SM. Hoyt and C.F. Abell. Variable, non-linear tie-line frequency bias for interconnected systems control, IEEE Trans. Power Syst., 3 (3) (1988). [2] Z.M. Ai-Hamouz and Y.L. Abdel-Magid, Variable structure load frequency controllers for multiarea power systems, Int. J. Electr. Power EnergJa Syst., 15 (5) (1993). [3] CC. Lee, Fuzzy logic in control systems: Fuzzy logic controller, parts I and II, IEEE Trans. Sq’st., Man Cybern.. 20 (2) (1990). [4] Zhen-Yu Zhao, Masayoshi Tomizuka and Satoru Isaka, Fuzzy
Systems
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42 (1997)
145-152
gain scheduling of a PID controllers, IEEE Trans. Syst., Man Cybern., 23 (5) (1993). J. Nanda, D.P. Kothari and D. Das, Discrete151 M.L. Kothari, mode automatic generation control of a two-area reheat thermal system with new area control error, IEEE Trans. Power Syst., 4 (2) (1989). [61 A. Kumar, O.P. Malik and G.S. Hope, Variable-structure-system control applied to AGC of an interconnected power system, IEE Proc., 132 Pt. C (I) (1985). M.L. Kothari and P.S. Satsangi, Automatic genera171 .I. Nanda, tion control of an interconnected hydrothermal system in continuous and discrete modes considering generation rate constraints, IEE Proc., 130 Pt. D, (1) (1983). PI D. Das, J. Nanda, M.L. Kothari and D.P. Kothari, Automatic generation control of a hydrothermal system with new area control error considering generation rate constraint, Electr. Mach. Power Syst., 18 (1990) 461-471. and M.H. Hamza, Comparison of three al[91 P. Agathoklis gorithms for load frequency control, Electr. Power Syst. Res., 7 (1984) 1655172. [lOI 0. P. Malik, A. Kumar and G.S. Hope, A load frequency control algorithm based on a generalized approach. IEEE Trans. Power Syst., 3 (2) (1988). [“I A. Rubaai and V. Udo. An adaptive control scheme for load-frequency control of multiarea power systems. Parts I and II. Electr. Power Syst. Res., 24 (1982). Regulation error in load frequency u-4 A. Bose and 1. Atiyyah, control, fEEE Trans. Power Appar. Syst., PAS-99 (2) (1980). G.S. Hope and O.P. Malik, Optimisation of 1131 S.C. Tripathy, load-frequency control parameters for power systems with reheat steam turbines and governor deadband nonlinearity, IEE Proc., 129 Pt. C, (1) (1982). and A. Marched, Dynamic behaviour of AGC u41 E.B. Shahrodi systems including the effects of nonlinearities, IEEE Trans. Power Syst.. PAS-104 (12) (1985). D.N. Ewart, L.H. Fink and A.G. USI N. Jaleey, L.S. VanSlyck, Hoffman, Understanding automatic generation control, IEEE Trans. Power Syst., PAS-89 (4) (1990). Lu, Chun-Chang Liu and Chi-Jui Wu, Effect of U61 Chun-Feng battery energy storage system on load frequency control considering governor deadband and generation rate constraint, IEEE Trans. Power Syst, IO (3) (1995).